where \(\kappa \)  is the Gauss curvature of \(M\) .
The form \(\omega (u)\)  is called the vorticity of \(u\) .
In the Euclidean setting, when \(u\colon \mathbb {R}^2\rightarrow \mathbb {R}^2\) ,
the vorticity \(\omega (u)\)  essentially reduces to the
Jacobian determinant of \(u\) , i.e. \(\omega (u) = 2(\det \nabla u) \, \mathrm {d}x\) .
As in the Euclidean case, if a sequence \(u_\varepsilon \)  satisfies
a logarithmic energy bound such as \(F_\varepsilon (u_\varepsilon ) \le C \left| \log \varepsilon  \right|\) 
(for some constant \(C\)  that does not depend on \(\varepsilon \) )
then the sequence of vorticities \(\omega (u_\varepsilon )\)  is compact.
Controlling the vorticity of \(u_\varepsilon \)  is key to
proving a sharp lower bound for the energy of \(u_\varepsilon \) 
and obtain compactness results for the minimizers of \(F_\varepsilon \) .
However, when the genus \(\mathfrak {g}= \frac{2-\chi (M)}{2}\ne 0\) , it is
also necessary to control the \(L^2\) -projection of the 1-form \(j(u_\varepsilon )\)  on the space of harmonic 1-forms, \(\mathrm {Harm}^1(M)\) .
\(\mathrm {Harm}^1(M)\)  is a real vector space of dimension \(2\mathfrak {g}\) .
Let \(\mathbb {P}\)  be the \(L^2\) -projection onto \(\mathrm {Harm}^1(M)\) .
Ignat and Jerrard [1]} showed that,
if a sequence of vector field \(u_\varepsilon \)  satisfies
suitable energy bounds, then there exist distinct points
\(a_1\) , ..., \(a_n\)  in \(M\) , integers \(d_1\) , ..., \(d_n\) ,
a form \(\xi \in \mathrm {Harm}^1(M)\)  and a (non-relabelled) subsequence,
such that
\(&\mathrm {d}j(u_\varepsilon )\xrightarrow{} 2\pi \sum _{j=1}^n d_j\delta _{a_j}\qquad \hbox{ in } W^{-1,p}(M) \quad \textrm {for any } p\in (1, \, 2) \\& \mathbb {P}j(u_\varepsilon ) \xrightarrow{}\xi \) 
